Case Study 2: Muonic Hydrogen and the Proton Radius Puzzle

Overview

In 2010, an international team of physicists measured the Lamb shift in muonic hydrogen — an exotic atom in which the electron is replaced by a muon — and obtained a proton charge radius 4% smaller than the value accepted from decades of electronic hydrogen spectroscopy and electron-proton scattering experiments. This discrepancy, known as the "proton radius puzzle," challenged the physics community for over a decade and forced a critical re-examination of both experimental techniques and theoretical calculations. This case study explores the physics of muonic hydrogen, the nature of the discrepancy, and its (partial) resolution.

Part 1: What Is Muonic Hydrogen?

The Muon: A Heavy Electron

The muon ($\mu^-$) is an elementary particle identical to the electron in all respects except mass: $m_\mu = 206.768 \, m_e = 105.658$ MeV/$c^2$. It carries the same charge ($-e$), the same spin ($1/2$), and the same electromagnetic interactions. The muon is unstable, decaying via the weak interaction:

$$\mu^- \to e^- + \bar{\nu}_e + \nu_\mu$$

with a mean lifetime of $\tau_\mu = 2.197$ $\mu$s. This is long enough for precision spectroscopy but short enough to require pulsed experimental techniques.

Muonic Hydrogen: A Scaled Atom

Muonic hydrogen ($\mu p$) is an atom consisting of a proton and a muon. Because the muon is 207 times heavier than the electron, all length scales in muonic hydrogen are compressed by a factor of $\sim 207$:

The key feature is the Bohr radius: $a_0^\mu = 256$ fm, compared to $a_0 = 52,900$ fm for electronic hydrogen. The muon orbits 200 times closer to the proton, making it far more sensitive to the proton's internal structure.

Why the Proton Radius Matters

The proton is not a point charge. It has a finite charge radius $r_p$, typically defined as the root-mean-square charge radius:

$$r_p^2 = \langle r^2 \rangle_{\text{charge}} = \int r^2 \rho_{\text{charge}}(r) \, d^3r$$

where $\rho_{\text{charge}}(r)$ is the proton's charge distribution, normalized so that $\int \rho \, d^3r = 1$.

The finite proton size shifts the energy levels of hydrogen. For an $s$-state ($l = 0$), the leading correction is:

$$\Delta E_{\text{size}} = \frac{2}{3}\frac{\alpha}{\pi} \left(\frac{Z\alpha m_r c}{\hbar}\right)^3 \frac{(\hbar c)^2}{m_r c^2} r_p^2 \cdot \frac{1}{n^3}$$

or more simply:

$$\Delta E_{\text{size}} = \frac{2\alpha^4 m_r^3 c^4 r_p^2}{3n^3 \hbar^2}$$

where $m_r$ is the reduced mass. Because this correction scales as $m_r^3$, it is $(m_\mu/m_e)^3 \approx 8.9 \times 10^6$ times larger in muonic hydrogen than in electronic hydrogen. This enormous enhancement is what makes muonic hydrogen the ideal system for measuring the proton radius.

Part 2: The Experiment

The CREMA Collaboration

The muonic hydrogen Lamb shift was measured by the Charge Radius Experiment with Muonic Atoms (CREMA) collaboration, led by Randolf Pohl (then at the Max Planck Institute for Quantum Optics in Garching, Germany). The experiment was performed at the Paul Scherrer Institute (PSI) in Villigen, Switzerland — one of the few facilities in the world capable of producing the required low-energy muon beams.

How to Make Muonic Hydrogen

1. Produce muons: A proton beam strikes a carbon target, producing pions ($\pi^-$), which decay into muons ($\mu^-$) with a known energy spectrum.

2. Slow the muons: The muons are decelerated to $\sim 5$ keV kinetic energy — slow enough to stop in a hydrogen gas target, but fast enough to reach the target before decaying.

3. Form muonic atoms: A slow muon entering hydrogen gas is captured by a proton into a high-$n$ orbit (typically $n \sim 14$) and quickly cascades down to the ground state, emitting X-rays at each step. About 99% of the muons reach the $1S$ ground state within $\sim 1$ $\mu$s.

4. About 1% end up in $2S$: A small fraction of muonic hydrogen atoms reach the metastable $2S$ state instead of the $1S$ ground state. These metastable atoms live for about 1 $\mu$s (limited by collisions with H$_2$ molecules, not by the intrinsic $2S$ lifetime).

The Measurement

The key measurement is the $2S \to 2P$ transition frequency (the Lamb shift). The CREMA experiment proceeds as follows:

1. A pulsed muon beam creates a burst of muonic hydrogen atoms.

2. After a delay of $\sim 0.9$ $\mu$s (allowing the non-metastable states to decay), a pulsed laser illuminates the target.

3. If the laser frequency matches the $2S_{1/2}^{F=1} \to 2P_{3/2}^{F=2}$ transition ($\lambda \approx 6$ $\mu$m), muonic atoms in the $2S$ state absorb a photon and transition to $2P$.

4. The $2P$ state immediately decays to $1S$ ($\tau \sim 8.5$ ps), emitting a 1.9-keV X-ray.

5. The X-rays are detected by large-area avalanche photodiodes surrounding the target.

6. By scanning the laser frequency and counting X-rays, the resonance is mapped out, yielding the transition frequency.

The Result

The CREMA collaboration reported their measurement in 2010 (published in Nature):

$$\nu(2S_{1/2}^{F=1} \to 2P_{3/2}^{F=2}) = 49,881.35(65) \text{ GHz}$$

From this transition frequency, using QED calculations of all other contributions to the Lamb shift, they extracted the proton charge radius:

$$r_p(\mu H) = 0.84184(67) \text{ fm}$$

This value was $4\%$ smaller than — and $5\sigma$ discrepant with — the CODATA 2010 recommended value of $r_p = 0.8775(51)$ fm, derived from electronic hydrogen spectroscopy and electron-proton scattering experiments.

Part 3: The Puzzle

Why 4% Matters

A 4% discrepancy in the proton radius may not sound dramatic, but the significance was enormous:

1. Statistical significance: The muonic measurement was 10 times more precise than the electronic measurements. The discrepancy was $5\sigma$ — far beyond the threshold for a fluctuation.

2. Implications for QED: If the muonic value were wrong, it would imply an error in the QED calculations that have been tested to extraordinary precision in other contexts.

3. Implications for scattering: If the electronic value were wrong, it would imply systematic errors in decades of electron-proton scattering experiments at facilities worldwide.

4. Possible new physics: Some theorists proposed that the discrepancy might signal new physics — a previously unknown interaction between muons and protons, mediated by a new particle. This would be the most exciting possibility.

Possible Explanations

The physics community explored several explanations:

1. Error in the muonic hydrogen experiment. This was considered unlikely given the clean resonance signal and the straightforward extraction of $r_p$ from the measured transition frequency. The CREMA collaboration subsequently confirmed their result with measurements of multiple transitions in muonic hydrogen and muonic deuterium.

2. Error in the QED theory for muonic hydrogen. The theoretical calculation of the muonic Lamb shift includes hundreds of QED diagrams, some computed only numerically. Multiple independent theoretical groups checked and confirmed the calculation, finding no significant errors. The proton polarizability contribution (the two-photon exchange) was identified as the largest theoretical uncertainty, but subsequent calculations confirmed it was not large enough to explain the discrepancy.

3. Error in electronic hydrogen spectroscopy or electron scattering. This emerged as the most likely explanation. The electronic hydrogen measurements of $r_p$ rely on the theoretical Rydberg constant, and the extraction of $r_p$ from scattering data involves model-dependent fits to form factors at low momentum transfer.

4. New physics: a muon-specific interaction. Several models were proposed, including exchange of a new dark photon ($Z'$) coupling preferentially to muons. However, these models were constrained by other experiments (muon anomalous magnetic moment, kaon decays, etc.) and no compelling model survived all constraints.

The Resolution (Partial)

Between 2017 and 2022, the puzzle was largely (though not entirely) resolved:

1. New electronic hydrogen measurements: The Hänsch group in Munich (2017) measured the $1S-3S$ transition in electronic hydrogen with unprecedented precision and obtained $r_p = 0.8335(95)$ fm — consistent with the muonic value, not the previous electronic value.

2. Revised $2S-4P$ measurement: A 2017 measurement by the Paris group of the $2S-4P$ transition also yielded a smaller $r_p = 0.8249(69)$ fm.

3. New electron scattering: The PRad experiment at Jefferson Lab (2019) used a new technique (a calorimetric detector) to measure $ep$ elastic scattering at very low momentum transfer, obtaining $r_p = 0.831(7)(12)$ fm — again consistent with the muonic value.

4. Reanalysis of older data: Several reanalyses of the older $ep$ scattering data, using improved fitting techniques, produced proton radii consistent with the muonic value.

As of 2024, the CODATA recommended value is:

$$r_p = 0.8414(19) \text{ fm}$$

which is essentially the muonic hydrogen value. The puzzle appears to have been caused by unrecognized systematic effects in the earlier electronic measurements, not by new physics. However, some tension remains, and the final resolution may require even more precise measurements.

Part 4: What We Learned

The Power of Muonic Atoms

Muonic hydrogen demonstrated that exotic atoms — atoms with one constituent replaced by a heavier particle — provide uniquely sensitive probes of nuclear structure. The enhancement factor of $(m_\mu/m_e)^3 \sim 10^7$ in the nuclear size correction makes muonic atoms the most precise tools for measuring nuclear charge radii.

The CREMA collaboration has since measured the Lamb shift in muonic deuterium, muonic helium-3, and muonic helium-4, extracting the charge radii of the deuteron, helion, and alpha particle with precision far surpassing any previous method.

The Importance of Redundancy

The proton radius puzzle highlighted the danger of relying on a single type of measurement. The electronic hydrogen value of $r_p$ was derived from a self-consistent analysis of many measurements, but those measurements all shared common theoretical inputs (the Rydberg constant) and similar systematic effects. The muonic hydrogen measurement used completely different physics and exposed previously hidden problems.

Theoretical Precision as a Discovery Tool

The resolution of the puzzle required theoretical calculations of extraordinary precision — hundreds of Feynman diagrams computed to multiple loops, nuclear structure corrections evaluated with modern few-body methods, and careful accounting of every contribution at the sub-MHz level. This theoretical infrastructure, built over decades for electronic hydrogen, was adapted and extended for muonic hydrogen, demonstrating the cumulative value of precision QED.

A Lesson in Humility

For over a decade, the physics community debated whether the proton radius puzzle required new physics. In the end, the most likely explanation was mundane: systematic errors in established experiments. This is a reminder that extraordinary claims (new fundamental interactions) require extraordinary evidence, and that careful, patient re-measurement of "known" quantities can be as valuable as exotic new experiments.

Discussion Questions

1. Why does the muon's heavier mass make muonic hydrogen more sensitive to the proton's charge radius? Derive the $(m_\mu/m_e)^3$ enhancement factor.

2. The CREMA experiment required a pulsed laser at 6 $\mu$m wavelength with enough energy to saturate the $2S \to 2P$ transition in a low-density gas. Estimate the required pulse energy given the transition's natural linewidth and the target geometry.

3. If the proton radius puzzle had been caused by a new muon-specific force mediated by a particle of mass $M$, what constraints on $M$ would the measured shift imply? Why did these models ultimately fail?

4. Compare the proton radius puzzle to other instances in physics where a discrepancy between theory and experiment was eventually resolved by re-measurement rather than new physics (e.g., the superluminal neutrino result from OPERA in 2011).

5. The CREMA collaboration plans to measure the Lamb shift in muonic lithium and muonic beryllium. What new physics would these measurements probe? What are the experimental challenges?

Further Exploration

• Read the original CREMA paper: Pohl, R. et al., "The size of the proton," Nature 466, 213-216 (2010).
• Read the review: Pohl, R., Gilman, R., Miller, G. A., & Pachucki, K., "Muonic Hydrogen and the Proton Radius Puzzle," Annual Review of Nuclear and Particle Science 63, 175-204 (2013).
• Explore the CODATA adjustments of fundamental constants (physics.nist.gov/constants) — the proton radius is one of the adjusted constants, and its value has shifted significantly between the 2014 and 2018 adjustments.
• Watch Randolf Pohl's talks on YouTube for an accessible overview of the experiment and its implications.